Revisiting this topic after a few years. I haven't read what I said on this topic from 3 years ago, but a lot has changed for me in that time.
I think one thing we can all agree on, is that there's nothing biologically wrong or deficient about a woman that prohibits her from being in the altar.
If it was a case of holiness or purity, then there are certainly many women out there far more worthy, holy & pure than many of us sinful men.
What I would say, and I'm surprised that no one has ever brought it up. Is that the altar is considered the Holy of Holies. It's the same as the Holy of Holies in the old temple. It's very sacred space that not even all men can enter, let alone anyone without a purpose or blessing.
The connection is important, because to our knowledge, there was only one woman who ever entered the Holy of Holies, and that was the Theotokos. Even the High Priests couldn't just enter without a rope being tied and bells on their garments for fear of dying.
In the same vein, we could argue that this is why women don't enter the altar typically. Not that they are prohibited (since monastic women and deaconesses do for a variety of reasons) from doing so. The Theotokos was the only woman who ever entered into the Holy of Holies, and she eventually bore the Son of God. It isn't that other women aren't "worthy" of it, but because their role in the church is different and because the Theotokos is just so special to us, that we honor her by keeping her and her son as the prime focus in the Holy of Holies.
Then there's the practical reason, which is the same reason why men & women were typically separated in churches. The Holy of Holies is such a holy and sacred space, and it's hard enough for the men in there to stay focused and pure of mind/thought that mixing genders together in the altar would invite the sorts of temptations and thoughts that arise out in the normal, everyday world. I think the church recognizes this, and that's why they've traditionally separated men from women, and one of the many reasons why women wear headcoverings. Yes we are in the world, but not of it, and when we are worshipping, we are not in the world, we are in the kingdom, and therefore, things that might be "okay" out in the world aren't okay in church because they can encourage thoughts or actions that are unholy, especially for the children of Christ standing at his throne worshipping him and receiving him.